Burnside's Lemma - Proof

Proof

The proof uses the orbit-stabilizer theorem and the fact that X is the disjoint union of the orbits:

\begin{align} \sum_{g \in G}|X^g| &= |\{(g,x)\in G\times X \mid g\cdot x = x\}| = \sum_{x \in X} |G_x| = \sum_{x \in X} \frac{|G|}{|Gx|} \\ &= |G| \sum_{x \in X}\frac{1}{|Gx|} = |G|\sum_{A\in X/G}\sum_{x\in A} \frac{1}{|A|} = |G| \sum_{A\in X/G} 1 \\ &= |G| \cdot |X/G|. \end{align}

Read more about this topic:  Burnside's Lemma

Famous quotes containing the word proof:

    “To cease to admire is a proof of deterioration.”
    —Charles Horton Cooley (1864–1929)

    “It comes to pass oft that a terrible oath, with a swaggering accent sharply twanged off, gives manhood more approbation than ever proof itself would have earned him.”
    —William Shakespeare (1564–1616)

    “Right and proof are two crutches for everything bent and crooked that limps along.”
    —Franz Grillparzer (1791–1872)